Section 5.3 Normal Distributions: Finding Values © 2012 Pearson Education, Inc. All rights reserved. 1 of 104

Section 5.3 Objectives Find a z-score given the area under the normal curve Transform a z-score to an x-value Find a specific data value of a normal distribution given the probability © 2012 Pearson Education, Inc. All rights reserved. 2 of 104

Finding values Given a Probability In section 5.2 we were given a normally distributed random variable x and we were asked to find a probability. In this section, we will be given a probability and we will be asked to find the value of the random variable x. xz probability © 2012 Pearson Education, Inc. All rights reserved. 3 of 104

Example: Finding a z-Score Given an Area Find the z-score that corresponds to a cumulative area of Solution: © 2012 Pearson Education, Inc. All rights reserved. 4 of 104

Solution: Finding a z-Score Given an Area Locate in the body of the Standard Normal Table. The values at the beginning of the corresponding row and at the top of the column give the z-score. The z-score is © 2012 Pearson Education, Inc. All rights reserved. 5 of 104

Example: Finding a z-Score Given an Area Find the z-score that has 10.75% of the distribution’s area to its right. Solution: © 2012 Pearson Education, Inc. All rights reserved. 6 of 104

Solution: Finding a z-Score Given an Area Locate in the body of the Standard Normal Table. The values at the beginning of the corresponding row and at the top of the column give the z-score. The z-score is © 2012 Pearson Education, Inc. All rights reserved. 7 of 104

Example: Finding a z-Score Given a Percentile Find the z-score that corresponds to P 5. Solution: The z-score that corresponds to P 5 is the same z-score that corresponds to an area of The areas closest to 0.05 in the table are (z =) and (z =). Because 0.05 is halfway between the two areas in the table, use the z-score that is halfway between and. The z-score is z 0 z 0.05 © 2012 Pearson Education, Inc. All rights reserved. 8 of 104

Transforming a z-Score to an x-Score To transform a standard z-score to a data value x in a given population, use the formula x = μ + zσ © 2012 Pearson Education, Inc. All rights reserved. 9 of 104

Example: Finding an x-Value A veterinarian records the weights of cats treated at a clinic. The weights are normally distributed, with a mean of 9 pounds and a standard deviation of 2 pounds. Find the weights x corresponding to z- scores of 1.96, -0.44, and 0. Solution: Use the formula x = μ + zσ z = 1.96: z = -0.44: z = 0: © 2012 Pearson Education, Inc. All rights reserved. 10 of 104

Example: Finding a Specific Data Value Scores for the California Peace Officer Standards and Training test are normally distributed, with a mean of 50 and a standard deviation of 10. An agency will only hire applicants with scores in the top 10%. What is the lowest score you can earn and still be eligible to be hired by the agency? Solution: © 2012 Pearson Education, Inc. All rights reserved. 11 of 104

Solution: Finding a Specific Data Value From the Standard Normal Table, the area closest to 0.9 is So the z-score that corresponds to an area of 0.9 is z = © 2012 Pearson Education, Inc. All rights reserved. 12 of 104

Solution: Finding a Specific Data Value Using the equation x = μ + zσ © 2012 Pearson Education, Inc. All rights reserved. 13 of 104

Section 5.3 Summary Found a z-score given the area under the normal curve Transformed a z-score to an x-value Found a specific data value of a normal distribution given the probability © 2012 Pearson Education, Inc. All rights reserved. 14 of 104